Sphere

 For other uses, see sphere (disambiguation).
Sphere.jpg
A sphere is, roughly speaking, a ballshaped object. In nonmathematical usage, the term sphere is often used for something "solid" (which mathematicians call ball). But in mathematics, sphere refers to the boundary of a ball, which is "hollow". This article deals with the mathematical concept of sphere.
Contents 
Definitions/Postulates
Great circle  The intersection of the sphere and a plane that contains the center of the sphere
 A great circle is finite and returns to its origional starting point
 There is a unique circle passing through any pair of nonpolar points
Polar points  The intersection of a sphere and a line passing through the origin of the sphere
 Polar points are opposite each other on a sphere
Arc of a Great circle  The shortest distance between two points on a sphere (while not going through the center)
Geometry
In threedimensional Euclidean geometry, a sphere is the set of points in R^{3} which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.
Equations
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In analytic geometry, a sphere with center (x_{0}, y_{0}, z_{0}) and radius r is the set of all points (x, y, z) such that
 <math>(x  x_0 )^2 + (y  y_0 )^2 + ( z  z_0 )^2 = r^2 \,<math>
The points on the sphere with radius r can be parametrized via
 <math> x = x_0 + r \sin \theta \; \cos \phi <math>
 <math> y = y_0 + r \sin \theta \; \sin \phi \qquad (0 \leq \theta \leq \pi \mbox{ and } \pi < \phi \leq \pi) \,<math>
 <math> z = z_0 + r \cos \theta \,<math>
(see also trigonometric functions and spherical coordinates).
A sphere of any radius centered at the origin is described by the following differential equation:
 <math> x \, dx + y \, dy + z \, dz = 0. <math>
This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other.
The surface area of a sphere of radius r is:
 <math>A = 4 \pi r^2 \,<math>
and its enclosed volume is:
 <math>V = \frac{4 \pi r^3}{3}<math>
The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area.
Einstein_gyro_gravity_probe_b.jpg
The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes.
A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.
Generalization to higher dimensions
Spheres can be generalized to higher dimensions. For any natural number n, an nsphere is the set of points in (n+1)dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number.
 a 0sphere is a pair of points <math>(r, r)<math>
 a 1sphere is a circle of radius r
 a 2sphere is an ordinary sphere
 a 3sphere is a sphere in 4dimensional Euclidean space
Spheres for n > 2 are sometimes called hyperspheres. The nsphere of unit radius centred at the origin is denoted S^{n} and is often referred to as "the" nsphere.
Generalization to metric spaces
More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set
 S(x;r) = { y ∈ E  d(x,y) = r } .
If the center is a distinguished point considered as origin of E, e.g. in a normed space, it is not mentionned in the definition and notation. The same applies for the radius if it is taken equal to one, i.e. in the case of a unit sphere.
In contrast to a ball, a sphere may be empty. For example, in Z^{n} with Euclidean metric, a sphere of radius r is nonempty only if r² can be written as sum of n squares of integers.
See also
Topology
In topology, an nsphere is defined as a space homeomorphic to the boundary of an (n+1)ball; thus, it is homeomorphic to the Euclidean nsphere described above under Geometry, but perhaps lacking its metric.
 a 0sphere is a pair of points with the discrete topology
 a 1sphere is a circle (up to homeomorphism); thus, for example, (the image of) any knot is a 1sphere
 a 2sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid is a 2sphere
The nsphere is denoted S^{n}. It is an example of a compact nmanifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere.
The HeineBorel theorem is used in a short proof that an nsphere is compact. The sphere is the inverse image of a onepoint set under the continuous function x. Therefore the sphere is closed. S^{n} is also bounded. Therefore it is compact.
See also
 Alexander horned sphere
 Ball (mathematics)
 Homology sphere
 Homotopy sphere
 Riemann sphere
 3sphere, hypersphere
External links
 Mathworld website (http://mathworld.wolfram.com/Hypersphere.html)de:Kugel
es:Esfera fr:Sphère gd:Baiscmheall he:כדור (גיאומטריה) ia:Sphera it:Sfera ja:球 ja:球面 nl:bol pl:sfera pt:Esfera ru:Сфера (поверхность) simple:Sphere sl:sfera fi:pallo sv:Sfär